BA have good reason for making very little effort to introduce any kind of bidding system to LHR WW fleet. The costs of building such a sytem are/were prohibitive due to the complexity of the current WW agreement.
It would have required a custom built program not one off the shelf used by another airline.
BA has wanted rid of said agreement for years. Spending any kind money on something you don't want or need would be pointless.
Of course, if the current WW fleet provided the savings themselves to fund the project then that's a different matter.